On reduction of differential inclusions and Lyapunov stability

Kamalapurkar, Rushikesh; Dixon, Warren E.; Teel, Andrew R.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 24.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is pointwise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory.

Classification : 93D02
Mots-clés : Differential inclusions, stability, hybrid systems, nonlinear systems

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